Abstract

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

Highlights

  • In this chapter, we are interested in the existence of solutions for the nonlinear fractional boundary value problem (BVP)8> Dα0þ uðtÞ þ f ðt, uðtÞÞ 1⁄4 0, t ∈ ð0, 1Þ,< >: uðiÞð0Þ 1⁄4 0, i ∈ f0, 1, 2, ... , n À 2g, Dβ0þ uð1Þ 1⁄4 Pp a jDβ0þ uη j: (1) j1⁄41We cover the multi-valued case of problem α 0 þ uð t Þ ∈ F ðt, uð t ÞÞ >

  • Inspired and motivated by the works mentioned above, we focus on the uniqueness of positive solutions for the nonlocal boundary value problem (1) with the iterative method and properties of f ðt, uÞ, explicit iterative sequences are given to approximate the solutions and the error estimations are given

  • We cover the multi-valued case of problem (2) when the right-hand side is nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler

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Summary

Introduction

We are interested in the existence of solutions for the nonlinear fractional boundary value problem (BVP). Iterative method is an important tool for solving linear and nonlinear Boundary Value Problems It has been used in the research areas of mathematics and several branches of science and other fields. In 2020 Li et al [41] consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions: 8> ÀCDq0þuÁðtÞ þ f ðt, uðtÞÞ 1⁄4 0,. In 2018, Bouteraa and Benaicha [10] studied the existence of solutions for the Caputo fractional differential inclusion cDαuðtÞ ∈ Fðt, uðtÞ, u0ðtÞÞ, t ∈ J 1⁄4 1⁄20, 1Š, subject to three-point boundary conditions. In 2019, Ahmad et al [43] investigated the existence of solutions for the boundary value problem of coupled Caputo (Liouville-Caputo) type fractional differential inclusions:. We give an example to ascertain the main result

Preliminaries
Existence results
Conclusions
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